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<h3 style="text-align:center;">Simulation of a Lennard-Jones system of particles</h3>

<p class="header_title">Introduction</p>

<p>The exact form of the interparticle potential u(r) for electrically neutral molecules and atoms has to be determined by a first principles quantum mechanical
calculation. Such a calculation is very difficult, and for many purposes it
is sufficient to choose a simple phenomenological form for
u(r). The most important features of
u(r) are a strong repulsion for small r and a weak attraction
at large r. The most common phenomenological form of u(r) is the Lennard-Jones or 6-12 proposed by John Edward Lennard-Jones in
1924:</p>
<p class="center">
<img src="lj.jpg" alt="" align="middle" >,
</p>
<p>The values of &#963; and &#949; for argon are &#963; =3.4 &#215; 10<sup>-10</sup> m and &#949; = 1.65 &#215; 10<sup>-21</sup> J.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The attractive
1/r<sup>6</sup> contribution to the Lennard-Jones potential is due to
the induced dipole-dipole interaction of two atoms.
Although each atom is electrically neutral, the instantaneous
fluctuations in the charge distribution can have nonspherical
symmetry. The resulting dipole in one atom can induce a dipole
moment in the other atom. The
resultant attractive potential is called the van der
Waals potential.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The repulsive interaction for small r is a consequence of the
Pauli exclusion principle. The electron wave functions of the
two molecules must distort to avoid overlap, causing some of the
electrons to be in different quantum states. The net effect is an
increase in kinetic energy and an effective repulsive interaction
between the electrons. The
1/r<sup>12</sup> form of the repulsive potential is chosen only for
convenience and is not derived from first principles.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The
existence of many calculations and simulation results for the
Lennard-Jones potential encourages us to consider it even though there
are more accurate forms of the interparticle potential for real
gas and liquids.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;It is possible to simulate a system of Lennard-Jones particles using either <a href="md/launcher.html">molecular dynamics</a> or <a href="mc/launcher.html">Monte Carlo methods.</p>

<p class="header_title">Problems</p>

<ol>
<li>Show that the minimum of the Lennard-Jones
potential is at
r<sub>min</sub> = 2<sup>1/6</sup> &#963; and that u(r<sub>min</sub>) = -&#949;.</li>

<li>At what value of
r is the force f(r) = du(r)/dr a minimum?</li>

<li> What is the value of the Lennard-Jones potential at r = 2.3 &#963;?</li>

</ol>

<p class = "small">Updated 10 May 2008.</p>

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